On images of subshifts under injective morphisms of symbolic varieties

TL;DR

This paper proves that the property of being a subshift of finite type or sofic is preserved under injective morphisms of symbolic algebraic varieties and modules over monoids, extending to cellular automata.

Contribution

It establishes the equivalence of subshift properties under injective morphisms for symbolic varieties and modules, generalizing previous results to broader algebraic structures.

Findings

Subshifts of finite type are preserved under injective morphisms.

Sofic subshifts are preserved under injective morphisms.

Results extend to admissible group cellular automata.

Abstract

We show that the image of a subshift $X$ under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, resp. a sofic subshift, if and only if so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be Artinian modules over a ring. We prove that for every closed subshift submodule $\Sigma \subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $\tau \colon \Sigma \to B^G$, a subshift $\Delta \subset \Sigma$ is of finite type, resp. sofic, if and only if so is the image $\tau(\Delta)$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.